1. Introduction
The vulnerability to natural disasters has increased all over the world during the past few years, leading to social, economic, and environmental tragedies. The tragic flooding in 2010–2011 in the state of Queensland, Australia, alone has resulted in over 200 000 people affected, with 35 confirmed deaths, as stated by BBC News (2011) and ABC News (2011). The resulting damage was over AUD $1 billion, with an estimated reduction of AUD $30 billion in gross domestic product (GDP). According to the Intergovernmental Panel on Climate Change, economic losses from weather- and climate-related disasters have increased in the past few decades, and heavy rainfalls associated with tropical cyclones (TCs) are likely to increase with continued warming (Allen et al. 2012).
Apart from the increase in the importance of disaster monitoring, there has been some decline in monitoring infrastructure as a result of disasters themselves (Asante et al. 2008). Millions of dollars are being spent on disaster management sensor networks, signifying the importance of efficient sensor deployment and management. One method of extending the lifetime of a wireless sensor network is to optimize the onboard energy consumption of nodes. To conserve onboard energy, many design approaches have been researched, such as network architecture, efficient sensing circuitry, algorithms, and communication protocols (Nanayakkara et al. 2011; Albers 2010). Various dynamic power management techniques have also been proposed, which mainly address sleeping patterns and idle states dynamically (Passos et al. 2005). In addition to power management in sensor nodes, efficiency and the effectiveness of a network can be improved drastically using the just-in-time sensor deployment, as described in Yee et al. (2006).
A predicted time of flooding for a given area prone to cyclone-induced flooding will immensely improve the effectiveness of disaster management operations. This includes power management techniques and dynamic deployment of sensor networks, which would help obtain a perfect balance between reducing network cost and capturing the most important data. It has been shown in Ding et al. (2010) that the losses in a catastrophic event, such as a natural disaster, decrease with the increase of predictability of the event. There are several deterministic flood models in use today, predicting likely inundations resulting from TC activity. A digital elevation model (DEM) of increasing resolution is used to model water flow (Asante et al. 2008; Bates and De Roo 2000; Bates et al. 2003), taking into consideration that flood inundation extent is highly dependent on topography (Bates and De Roo 2000). It has been shown in Bates and De Roo (2000) that the predictive ability deteriorates with the decrease in resolution of DEM data used. This was especially true when levee structures were smoothed with lower-resolution landscapes. However, it has also been shown that the improvement of prediction accuracy with increased resolution is marginal for smoother landscapes. Therefore, one drawback of this approach is the fact that it uses high-resolution data throughout the landscape, which results in increased computational cost. This raises the importance of using different resolutions based on the geography of the area. To address this issue and to improve prediction efficiency through offline calculations, this paper introduces the concept of geographic primitives (GPs). A GP can be defined as a portion of landscape where the main driving force of a disaster shows statistically stereotypical behavior (e.g., distinguishable water flow patterns can define GPs in the process of flood prediction, where a few identifiable GPs would be basins, mountain ranges, valleys, flat land, etc.). Identifying GPs is done through visual inspection of contour patterns that would give low modal vector fields with patterns typical to the GP type. A detailed discussion comparing the concept of GPs with other methods used for spatial discretization is included in section 1 of the supplemental information.
Models have also been developed to incorporate data assimilation (Bates et al. 2003; Zerger and Wealands 2004). In Bates et al. (2003), the authors have addressed the problem of obtaining a topographically optimum model to improve the representation of “raw” topographic data so that its integration with lower-resolution numerical inundation models is optimal. Zerger and Wealands (2004) integrate flood model outputs with a Geographical Information System (GIS). A model currently in use for streamflow monitoring is the Geospatial Stream Flow Model (GeoSFM), which is a semidistributed hydrological model developed as an extension of the ArcView GIS software (Asante et al. 2007). GeoSFM software uses a wide range of inputs, including satellite rainfall estimates, soil data, land cover, and elevation data, to predict streamflow. One shortcoming of this model is its inability to predict absolute flow magnitudes because of the absence of regional and seasonal bias correction (Asante et al. 2008), which is a difficult task with data limitations when trying to use a generic model for diverse regions. The GeoSFM model only finds abnormalities in water flow. One other limitation is that the output is numerical and the probability aspects are not present in the output.
Large file size problems for high-resolution topography can be addressed by dividing the landscape into multiple primitives. This division could be a regular grid or an irregular division, as suggested in the deployment of disaster management wireless sensor networks in Da Silva et al. (2010). We propose the concept of GPs is a physically meaningful strategy to simplify calculations. Figure 1 shows some identified geographic primitives on the landscape of Queensland, leading to some stereotypical flood distributions including scattering, converging, and diverging. As topography will not always give clearly distinguishable primitives, catchments need not have strict boundaries, and using overlapping boundaries is more appropriate. It should also be noted that the runoff between certain GPs would show stereotypical behavior.
The lead time for TC-induced flood prediction can be further increased by predicting TC-induced rainfall, which results in floods. Although there have been great advancements in numerical prediction models in the past decade (Lonfat et al. 2004, 2007; Ebert et al. 2010), these models still have room for improvement, owing to the multitude of factors affecting TC-induced rainfall; getting data on all these factors in real time is not practical, but a probabilistic approach would be more viable.
The Bayesian probability approach has been increasingly used in natural disaster prediction (Ding et al. 2010; Li et al. 2010; Lu et al. 2010) because of the many advantages and flexibility associated with it. One main advantage in the approach is that it incorporates prior knowledge, pragmatically optimized by the user, which allows for probabilistic predictions as opposed to binary true–false outcomes, which have a risk of misleading forecasts. In addition, this approach accounts for parameter uncertainty, reducing the error from the overfitting of training data, and provides a natural interpretation of regularization (Bermak and Belhouari 2006). Therefore, given the large amount of factors affecting cyclone-induced rainfall characteristics (Liu et al. 2012), the possibility of errors in observed data, and the ability to use prior knowledge in prediction (Madsen and Jakobsen 2004), using a Bayesian framework can be suggested as a suitable candidate for TC-induced rainfall prediction.
Including the prediction of the path of a cyclone could further increase the lead time of flood prediction by the model. Cyclone track prediction models have come a long way since the use of purely statistical models such as the Climatology and Persistence (CLIPER) model, proposed by Bessafi et al. (2002), which is now used solely as a benchmark for assessing the skill of other models. Dynamic models, which use numerical weather prediction, are the most widely used at present. These models generally require supercomputers to solve the mathematical equations governing the physics of the atmosphere and use numerical methods to solve these equations in order to generate forward-in-time forecasts of the track of the cyclone (University of Rhode Island 2012). The Geophysical Fluid Dynamics Laboratory (GFDL) model (Bender et al. 2007) is one of the most widely used dynamic models. The dynamic models use a large range of data sources for assimilation, including satellite data, specialized aircraft data, and local area sensor networks. Gall et al. (2011) states that GFDL is a regional model as well a “late” model, where the first prediction is only available 4–6 h after the initial track advisory is released, though it is high in accuracy. Considering these limitations, the output of this model could be used for the flood prediction model.
The rest of this paper is organized as follows. In section 2, we describe the proposed model, expanding on all steps. Section 3 shows simulations of the model using real data from Cyclone Yasi and Cyclone Tasha, which made landfall in early February 2011 and late December 2010, respectively, in Queensland, Australia. Simulation results of the above are discussed in section 4, and section 5 gives the concluding remarks of the paper.
2. Proposed model for flood prediction
The purpose of the model is to calculate the earliest possible prediction of the expected time of flooding resulting from cyclone-induced rain for a given geographical area. This knowledge is to be used in optimizing disaster management operations and disaster management sensor networks.
Consider a geographical area, prone to TCs, with a height profile h(x_{1}, x_{2}), where x_{1} and x_{2} describe the location in terms of latitudinal and longitudinal coordinates, respectively. Prediction starts with available minimal data of the cyclone while it is refined with incoming data. Let t = T, 2T, 3T, … , kT be prediction times for k number of periodic predictions at T time intervals. Let the geographical area consist of many GPs, where the propagation of floods show statistically stereotypical behavior (e.g., basins, mountain ranges, valleys, and flat land). Let Π be the set of node clusters with data currently available and Π′ be those which are yet to sense any data. Prediction of failure time (i.e., time to flood) for node i with data FT_{i∈Π}, as well as for nodes in Π′, start as soon as cyclone data is sensed anywhere in the total network. This data is used initially for cyclone path prediction, where the output is used in the rainfall density distribution prediction phase, and finally into flood density distribution prediction, as explained in Fig. 2.
Any subsequent data (cyclone path and rainfall) is used in refining the prediction as data becomes available. The failure time for the nodes in clusters (i ∈ Π′) can be inferred from available data (i ∈ Π). This estimate can be recalculated as more data is obtained, and the FT_{i} estimate at a given time can be used to deploy and manage the ith node cluster. This is possible even when there is no current available data at any of the nodes (i.e., Π ∈ ⊘), where the prior value is mainly decided by historical data and mathematical simulations of water flow. The summary of the symbols used in describing the model are listed in Table 1.
Parameters used.
The basic sections under the development of the proposed model are given in Fig. 2. Prediction starts with incoming cyclone best-track data, which are used for cyclone path prediction to get a linear approximation of the path using a sliding window of samples where the window size and deviation depends on the speed of the eye of the cyclone. The output is used as an input to the rainfall prediction phase. Probability distribution of rainfall is predicted using Bayesian learning, with the initial prior value taken from the rainfall (R)-CLIPER model and a likelihood function generated using the available rainfall data. The rainfall prediction is used for the next stage for flood prediction by considering the dynamics of water flow on a landscape approximated by a linear combination of Gaussian functions. The area is segmented into geographic primitives identified by water flow patterns during the learning phase, and the calculation of time to failure is made more efficient with the use of probability transition matrices for each segment. Figure 2 gives the flowchart of prediction steps, which are discussed in detail in the following subsections.
Several assumptions were made in developing this model, mainly for simplicity and to focus more on the new concepts introduced in the paper. The main assumptions are 1) that the terrain is frictionless, 2) that soil absorption is negligible, 3) that evapotranspiration during prediction time is negligible, and 4) that topography of the area remains unchanged during the time of prediction. Although the first three effects are out of the scope of this paper, they can be readily integrated into the model in the motion equations and probability transition matrices, which are described in detail later in the paper, and thereby improve the accuracy. Assumption 4 is the basis of the proposed offline calculations.
a. Predict cyclone path with available data
b. Predict rainfall density distribution with available data
c. Predict water deposit density distribution with available data
The next step in the proposed model is to calculate flood distribution through water flow calculations using the predicted rainfall distribution as input.
1) Water flow calculations
In the proposed model, water flow calculations were carried out in latitudinal and longitudinal directions. In this model, it is assumed that the rainwater would go down a slope at a constant velocity between two sample times. The algorithm used for calculating water flow for one direction in one subportion of a time period is given in Algorithm 2 in the appendix, and the initial simulation steps used in developing the model are described in section 4 of the Supplement.
2) Gaussian approximation of real landscape
The landscape of Queensland, Australia, is used in the simulations of the proposed model. The area was simulated using 250-m data from the Shuttle Radar Topography Mission (SRTM) (CGIAR-CSI 2011), and resolution was coarsened to 25 km by taking the simple arithmetic mean. It should be noted that the model would give better results with the original higher-resolution data if computational resources were not a limitation. The landscape of Queensland was reconstructed using a bootstrapping technique to approximate it with a linear combination of bivariate Gaussian functions with orientation. The reasons for this step are that the landscape was required to be approximated by a differentiable function in order to be used in Eqs. (6) and (7), to generate unlimited offline data from limited available topography data, to remove errors in calculation resulting from the roughness of raw data, and to reduce computational costs associated with large datasets when dealing with numerical gradients. The approximation was done with an initial equally spaced matrix of Gaussian functions, the iterative minimization of a cost function dependent on error, and the adjustment height and angle variables as explained in the steps below.
Although using a Gaussian-approximated function for landscape reduces the limitations of using numerical gradients in calculations, the water flow calculations for any given landscape take up a considerable amount of processing since the water displacements need to be calculated for each sampling period and for each coordinate point on the landscape. This is especially true when the area of the landscape considered is large and would not be efficient on a real-time system. Using the fact that dynamics of water would not change with time for the given landscape, the products of the calculations are stored in a matrix, which is used to get the probability distribution of water accumulation at any future referencing time, given the initial rainfall distribution probability.
3) Using a probability transition matrix to improve calculation efficiency
3. Simulation results
Here we compare the proposed model with two datasets from Cyclone Yasi and Cyclone Tasha, which made landfall in Queensland, Australia, in January–February 2011 and December 2010, respectively. We discuss the simulations of Cyclone Yasi in detail, while we only include a few simulation steps with data from Cyclone Tasha. The landscape of Queensland was taken as the considered land with x_{1} (latitude) ranging from 30°S to 10°S and the x_{2} (longitude) between 135°E and 155°E. The velocity of water flow was taken to be υ = 0.25 m s^{−1} and computing time interval as T = 15 min = 900 s. The radius of the earth was taken as 6 378 100 m. The topography of Queensland was modeled using the coarsened SRTM terrain data, as previously described. The 48 × 48 Gaussian functions were linearly combined to approximate the landscape of Queensland as described in section 2, setting i = 1, 2, … , 48 and j = 1, 2, … , 48 in Eq. (9). The landscape plotted from the SRTM data is compared to the landscape simulated using the above method in Fig. 3.
The rainfall data for the time period under Yasi’s effect was obtained from NASA’s Goddard Earth Sciences (GES) Data and Information Services Center (DISC) Interactive Online Visualization and Analysis Infrastructure (GES DISC 2011). The best-track data of Cyclone Yasi, which entered Queensland, Australia, in January–February 2011, was obtained from the Regional and Mesoscale Meteorology Branch (RAMMB 2011) and is plotted in Fig. 4. The same data was obtained for Cyclone Tasha, which entered Queensland on 24 December 2010.
Figure 5 shows the predicted path, with 95% confidence bounds, of Cyclone Yasi with incoming data, using the prediction steps, with a moving window, described earlier. The four graphs show how the prediction changes with incoming data and the use of the moving window and compares it with the actual track of the cyclone. The allowable deviation is calculated taking the speed of the cyclone into consideration, and the window size for prediction is dependent on the R^{2} value of linear fit and the speed, as discussed under the proposed model. An animation of path predictions is included in the Supplement (file E1.mpg), and Table 2 compares the official 12-h track forecast from the Joint Typhoon Warning Center (JTWC) advisories to the proposed model’s track forecast. A similar simulation for Cyclone Tasha is shown in Fig. 6.
Comparison of errors (to the nearest kilometer) of official 12-h cyclone track forecasts with the proposed model’s forecasts.
The next step of the process was predicting rainfall distribution resulting from the cyclone. In Fig. 7, the output of the proposed model is compared with actual rain gauge data for Queensland in January–February 2011. We compare this with the predictions using the R-CLIPER model and using the same Bayesian inference model proposed, with a bivariate Gaussian likelihood function without orientation. The equations for the R-CLIPER model from Marks et al. (2002) are given in section 3 of the Supplement. The error for each step of the rainfall prediction stage, as shown in Fig. 7, is calculated as the distance of the prediction mean’s contour centroid from that of the mean rainfall. The same comparison for Cyclone Tasha is shown in Fig. 8, which made landfall in Queensland, Australia, in late December 2010.
Initially, the landscape was divided into overlapping subareas of dimensions 3° latitude by 3° longitude, with overlapping of 0.833° in each direction. This was done for simplicity in simulations. Flood distribution with time was simulated using a PTM for each of these subareas. File E2.mpg, included in the Supplement, is an animation showing how the water deposit distribution changes with time for particular subareas (in all figures, the water deposit distributions are scaled up to be visually comparable with height). More details of the simulation steps followed are described in the Supplement. The fact that accuracy of prediction depends on the GP type is elaborated in Fig. 9. The GP is a valley in the top two figures, where the flood prediction is less affected by a small error in the rainfall distribution approximation. In contrast, the error in flood prediction stage is magnified for the terrain in the bottom two figures where the GP is composed of a mountain range dividing the area into two portions. Therefore, in the output prediction, the variance of the output is highly dependent on the GP type. The topographic contour plots of three subareas with the contour plots of their PTMs are given in Fig. 10, illustrating the PTM’s dependency on landscape. The absorbing states of the PTM of the landscape in Fig. 10c show that the valley is an absorbing state for water deposit distributions. Therefore, it is evident that the properties of the topography in a given area can be identified mathematically through its probability transition matrix and its properties. This is especially important when the landscape is complex and the PTM can be used to identify GPs.
The same simulations were run on identified GPs of irregular shapes. Two such GPs are plotted in Fig. 11. First, GP is a valley that can be treated as an independent portion of calculations where water would neither enter nor leave the GP. The absorbing states plotted as dots show that the bottom of the valley is an absorbing state, and there are two such minimum points in this subarea. Figures 11c,d show another GP of one side of a mountain range. There are no absorbing states in this PTM, and water will only flow out of it and never into it. Therefore, in both these GPs, we could run water flow calculations, ignoring the inflow term in Eq. (15). An animation of the water deposit distribution with time for the former GP is included in the Supplement (file E3.mp4).
4. Discussion
Numerous uncertain environmental factors shaping the formation, development, and propagation of a cyclone pose complex computational challenges to predict the damage caused after its landfall. We have presented a distributed Bayesian framework defined across a given set of geographical primitives to integrate prior knowledge with real-time measurements to make accurate and timely predictions. Moreover, it allows starting flood prediction as soon as sufficient data points of the cyclone path are received. This would be before the cyclone makes landfall, allowing preparation well in advance. The prediction gets more and more refined with any incoming data. Furthermore, the partitioned approach allows asynchronous data to be input at any stage of the computation.
In the rainfall prediction stage of the model, we have used all available rainfall data and not used any other features such as wind shear or topology. The results summarized in Table 3 show a 33% increase of accuracy for rainfall density distribution prediction, suggesting that, by using Bayesian inference technique with recurrent learning, the prediction could be better than with methods using more sophisticated data assimilation. The main reason for the increase in accuracy is the ability of the proposed Bayesian learning–based method to combine the advantages of data-driven and heuristic modeling by amalgamating past knowledge through the prior distribution and new data through the likelihood function.
Performance summary of major steps of the proposed model.
The effectiveness of using a simple approach with minimal features is also apparent in the short-term cyclone path prediction approach included in the proposed model, where the comparison results in Table 2 show that the model forecasts are better for most cases, while noting that official forecasts are released 3 h later than forecasts of models because of the need to collect a large amount of data from many sources.
A main feature of the model is the use of GPs, which encapsulate the dynamics of water flow on static landscape primitives. Through statistically summarizing simultaneous events that are spread across geography, it allows us to have a distributed updated algorithm that leads to parallel computation. Clustering of these primitives permits better understanding of the distributed nature of flood propagation and its relation to any geographical feature. The speed of calculations increases by almost 20 times with the use of PTM on GPs, as opposed to using gradient-based calculations for each prediction time step. At this point, we use manual demarcation of GPs in order to simplify the computational process. However, this could be extended to mathematically identify GPs, for example, by using numerical methods such as the Delaunay triangulation (Lee and Schachter 1980). This is outside the scope of this paper but is included in our future work. The efficiency increases further when predicting for multiple time steps using Markov chains. All comparisons were done using an Intel® Core^{TM} i7 CPU Q 740 @ 1.73 GHz processor with 16 GB available memory. The accuracy was reduced by the resolution of the states of the PTM, which resulted in a maximum possible error of 6 km (at transition states of the PTM) with the resolution used in simulations. This reduces further with time as the PTM approaches absorbing states, where water would be stagnated, resulting in floods. The significance of this error is further reduced with the use of variable resolutions considering the type of GP (e.g., using higher resolution in valleys and lower resolution on flat land).
In addition to effective and accurate prediction, the proposed stochastic process for identifying GPs with overlapping boundaries, using complex intrinsic statistics of flood propagation, allows mathematical identification of areas with flood risk even before using any rain data. This knowledge can be used in optimizing disaster management strategies, including land use. The terrain approximation provides a differentiable function, permitting the development of a Markov chain process based on simulation data that bootstraps the real-time computation based on measured rainfall data.
Despite the mentioned advantages, there are a few limitations of the model which could be addressed in future work. One considerable limitation is that, in the rainfall prediction step, projection is limited to an area of 500 km around the eye of the cyclone. This neglects the resulting rainfall due to convergence along the coasts during extratropical transition (ET) and accumulation of rainfall in rainband echoes over specific regions farther away from the eye of the cyclone, as pointed by Lonfat et al. (2007). Additionally, although simplicity has its advantages, the model could be further improved by including some of the most significant features affecting cyclone activity, which will have less effect on efficiency. A more complete treatment of hydrologic aspects can make the predictions more generic by relaxing some of the assumptions such as frictionless and impermeable surfaces. If there are already existing relevant datasets for the area, they can be readily used to get a better flood vector field for the use in demarcation of GPs. It is still worth noting that, although such comprehensive approaches make the prediction more realistic, great care has to be taken to get high-quality data since the accuracy of prediction is highly sensitive to the used data. However, one important feature in using Bayesian updating is that it is purely data-driven, and therefore, for the rainfall prediction step, any error arising from the initial assumptions is suppressed after a few calculating time steps.
It was also noticed that, for the cyclone path prediction step, while the model’s 12-h prediction was indicated to be better, the 72-h prediction of the JTWC advisory was generally better than the 72-h prediction of the proposed model. Therefore, the model’s prediction could be further improved using these outputs as input to the model using its modularized structure. Meanwhile, it should be noted that the model produces the predictions 3 h before the official forecasts are available, which would be much helpful for just-in-time sensor deployment and management, and the JTWC advisories would serve only to improve on that prediction. The model’s flood prediction does not consider existing water storage in the calculations. This could be included simply by considering the water levels of existing storage as a part of the landscape and building the Gaussian approximation of land from it. The flood levels can be defined by reducing the maximum capacities of water storage.
The proposed model has only been validated with datasets of two cyclones, both of which made landfall in the same state of Australia. But it is also worth noting that the mathematical framework presented in this paper does not use any assumptions specific to Queensland. Our proposal is a probabilistic model of which the output is dependent solely on data. Therefore, datasets of any cyclone in any region could be used with the proposed model without further modification.
5. Conclusions
A distributed Bayesian framework, defined across a set of predefined geographical primitives, is developed to predict the time to flood resulting from TC-induced rainfall. The purpose of our new model is to integrate prior knowledge with real-time measurements in order to produce accurate and timely flood predictions to be used in planning disaster response operations, including the deployment and management of wireless sensor networks. The flexibility in data assimilation resulting from the model’s modularized structure is an important attribute of the proposed model. Another advantage of the model is that prediction starts well ahead of time, with minimal available data, and this prediction is refined with any subsequently available data. The numerical simulation results suggest that the proposed model outperforms alternative probabilistic and numerical models that use multiple sources of data in terms of efficiency and accuracy for short-term forecasts. Further, the modularized and data-driven approach of the proposed model allows integrating prediction outputs available from any other models in subsequent predictions, which could increase the accuracy of long-term predictions of the model.
We have presented the model’s performance on the data from Cyclone Yasi, which made landfall in Queensland in early 2011. The model proves to be relatively high in efficiency, resulting from the fact that there are numerous factors that affect the formation, development, and propagation of cyclones, and the more complicated models that use more features need time to collect the data, which in turn results in delays in prediction. However, these complex models do not show significant improvements in accuracy, as the used features are still limited. The results summarized in Table 3 show that the proposed model produced cyclone path predictions up to 3 h ahead of official cyclone path forecasts in the 12-h horizon with a 6.5% improvement of accuracy. The accuracy of rainfall prediction improved by 33.7% compared to the R-CLIPER model with a mere 10% (in the order of a few seconds) increase of computing time. Finally, the flood prediction stage gave a considerable 18.9 times higher processing speed compared to using a gradient-based method for any given resolution.
In conclusion, we propose that the concept of using nonuniform geographical primitives with known statistical features for propagating a given disaster could be more generally used to improve the accuracy of estimating the time to pass a critical stage at a given place for other types of disasters, such as bush fires. For instance, in a bush fire scenario, the selection of GPs would be dependent on statistical features of wind direction in a given time frame.
Acknowledgments
This research was partly supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC) Grants EP/I028765/1 and EP/I028773/1.
APPENDIX
Algorithms Used in Proposed Model
a. Algorithm used for cyclone path prediction
- Algorithm 1: Simulate cyclone path
- Step 1: Set i = 0;
- Step 2: Collect ith location data input until i = n, where n is the default size of the moving window used for calculations;
- Step 3: Calculate the current speed of the cyclone and adjust window size for Step 4 accordingly (a default step size of 5 is used in the calculation, which reduces to 3 when two consecutive readings show 30% below average speed and further reduces to 2 when three or more closest readings show 30% below average speed;
- Step 4: Use linear regression from Eqs. (2) and (3) to get a and b;
- Step 5: Calculate upper and lower margins of the distribution using the standard deviation weighted by the speed of travel of the eye of the cyclone and the R^{2} value of fit;
- Step 6: Continue at Step 3 for each received location data point.
b. Algorithm used in calculating water flow
- Algorithm 2: Simulate water flow
- for time ∈ {t_{1}, … , t_{m}} do
- if |Current Slope − Last Slope| > |Current Slope| and
- |Current Slope − Last Slope| > |Last Slope| then
- Accumulate
- end if
- if Slope at x < 0 then
- end if
- if Slope at x > 0 then
- end if
- end for
Here, the T time period is broken down to m number of iterations for increased accuracy.
c. Algorithm used in approximating the landscape with a summation of bivariate Gaussian functions with orientations
- Algorithm 3: Approximate Landscape
- i_{t} = 0
- while
do {max no. of iterations = } - i_{t} = i_{t} + 1
- for each coordinate point i, j do
- error = h_{actual} − h_{c}
- if error ≤ ε then {check stopping criterion}
- break;
- end if
- J = (1/2)e^{2}
- end for
- end while
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